Introduction to Absolute Value Functions (HSC SSCE Mathematics Advanced): Revision Notes

Introduction to Absolute Value Functions

What is absolute value?

Absolute value refers to the distance of a number from zero on a number line. It tells us the magnitude or size of a real number, without considering whether the number is positive or negative. This means the absolute value is always non-negative.

For example:

• The absolute value of $5$ is $5$ (it's $5$ units from zero)
• The absolute value of $-5$ is also $5$ (it's still $5$ units from zero)
• The absolute value of $0$ is $0$ (it's at zero)

The absolute value function

The absolute value function is written as:

$f(x) = |x|$

where:

• $x$ is a real number input
• $f(x)$ is the absolute value of $x$, which is always non-negative

This function creates a relationship between all real numbers (inputs) and non-negative real numbers (outputs). Every real number is mapped to exactly one non-negative output value.

Piecewise definition

The absolute value function can be expressed as a piecewise function, meaning it has different rules depending on the value of the input:

$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$

This definition tells us:

• When the input is zero or positive ($x \geq 0$), the absolute value equals the input itself
• When the input is negative ($x < 0$), the absolute value equals the opposite of the input (which makes it positive)

For instance:

• If $x = 7$, then $|x| = 7$ (since $7 \geq 0$, we use the first rule)
• If $x = -7$, then $|x| = -(-7) = 7$ (since $-7 < 0$, we use the second rule)

Domain and range

The domain of the absolute value function is all real numbers: $(-\infty, \infty)$

This means we can input any real number into the function.

The range of the absolute value function is all non-negative real numbers: $[0, \infty)$

This means the output will always be zero or positive, never negative.

Graphical representation

The graph of $y = |x|$ creates a distinctive V-shaped curve with its vertex (turning point) located at the origin $(0, 0)$.

Key features of the graph:

• The graph is symmetric about the y-axis, which means it has even symmetry. If you fold the graph along the y-axis, both halves match perfectly
• For $x \geq 0$, the graph follows the line $y = x$ (ascending with slope $1$)
• For $x < 0$, the graph follows the line $y = -x$ (also ascending with slope $1$, but reflected)
• The lowest point on the graph is $(0, 0)$, and the graph extends upward indefinitely in both directions

Using absolute value to model distance

Absolute value equations are particularly useful for representing distance problems. Since distance is always measured as a positive quantity, absolute value is the perfect tool.

Worked example: Writing distance equations

Worked example: Evaluating using the piecewise definition

Worked example: Real-world application

Key concept: The absolute value function $f(x) = |x|$ maps real numbers to non-negative outputs. It can be defined piecewise as $x$ when $x \geq 0$, or $-x$ when $x < 0$.

Square root and absolute value relationship

There is an important mathematical identity connecting square roots and absolute values:

$\sqrt{x^2} = |x|$

where $x$ is a real number input.

This identity works because both expressions always yield a non-negative result, which matches the absolute value's definition.

Let's verify this with some numerical examples:

• For $x = 3$: $\sqrt{3^2} = \sqrt{9} = 3 = |3|$ ✓
• For $x = -3$: $\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|$ ✓
• For $x = 0$: $\sqrt{0^2} = \sqrt{0} = 0 = |0|$ ✓

This relationship holds because the square root function returns the non-negative root, which aligns perfectly with how absolute value works. Squaring a number makes it positive (or zero), and then taking the square root gives us the non-negative value, which is exactly what absolute value does.

Worked example: Verifying and applying the identity

Key concept: The identity $\sqrt{x^2} = |x|$ holds because both expressions yield non-negative equal outputs. Use numerical substitutions to verify, and apply this identity to simplify expressions involving square roots of squared terms.

Remember!